Dold and Lashof compare their construction for a monoid M to Milnor's when M is a group G. They give an explicit comparison for the first stage of the constructions. Somewhere I've seen the general n-stage comparison formula - anyone remember where?
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$\begingroup$ For groupoids, there were Buffet, Jean-Philippe; Lor, Jean-Claude Une construction d'un universal pour une classe assez large de Γ-structures.(French) C. R. Acad. Sci. Paris Sér. A-B270(1970), A640–A642; and Homotopy and integrability. Haefliger, André Lecture Notes in Mathematics, Vol. 197 Springer, Berlin, 1971, pp. 133–163. $\endgroup$– Gael MeigniezCommented Jul 12, 2023 at 23:40
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$\begingroup$ Thanks - several relevant papers have been `forgotten'' $\endgroup$– jim stasheffCommented Jul 14, 2023 at 0:18
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